Soft Selection: Why it is Multilevel Selection

It has come to my attention that it makes sense to spend a blog entry talking strictly about contextual analysis and soft selection. The problem which, as Okasha (2006 Evolution and the levels of selection) puts it, some “theorists find deeply counter-intuitive” is that in soft selection every group puts out exactly the same number of offspring individuals each generation. As a result there can be no variance in fitness among groups, and yet, using contextual analysis one would conclude that despite (and actually because of) this lack of variation in fitness among groups that there must be group selection acting. As evidence I can stand up a few of my favorite straw men, West, Griffin and Gardner (2007, J. Evol Biol 21:374-385, p. 380):

However, now consider that, because of localized resource competition, all groups have a fixed productivity (soft selection; Wallace, 1968) and all competition for reproductive success occurs within the group. . . . Contextual analysis therefore identifies both an impact of individual-genotype and also an impact of group-genotype on the individual’s fitness, and hence diagnoses the operation of both individual and group selection. Again, this is undesirable, as group selection should not be in operation when all groups have the same fitness.

I have to mention they earn full credit for cluelessness, which is immediately obvious in this quote with the discussion of “individual” and “group” genotype. (Um, we are phenotype view evolutionary biologists. Get with the program guys.), but also because the quote is from a section titled “There is no formal theory of group selection”. (Dang, and all this time I thought quantitative genetics WAS a formal theory.)

Clueless 2

Clueless: A classic movie that, of course, is not relevant to this weeks post.  (Hey, diss group selection, prepare to be dissed back)

Soft selection is one of a selection scheme put forward by Wallace (1968 Polymorphism, population size, and genetic load. In Population biology and evolution. Lewontin RC (ed.), pp. 87-108) in which there is a set of populations. At the end of each reproductive cycle each population produces the same number of migrants, which are the winners of individual selection acting within the population.

soft selection schmatic

Soft selection: There is individual selection within each population for a fixed number of migrants that will be produced at the end of the reproductive cycle. This is contrasted with hard selection in which individual fitness is unconstrained by group membership. (http://pedrovale.wordpress.com/2013/07/08/killing-them-softly-managing-pathogen-polymorphism-and-virulence-in-spatially-variable-environments/)

The point is, because there is no variation in output of the different groups, there is by definition no variation in mean group fitness, and as a result it is reasoned that there is no group selection. Indeed this is the conclusion reached by Wade (1985 Am Natur. 125, 61-73), in which he states “For soft selection this covariance [between group mean relative fitness and group mean phenotype] is zero by assumption. Even if the within-group genotypic fitnesses, Wij, were frequency dependent, the assumption [of constant mean group fitnesses] would prohibit the operation of group selection.” I doctored the quote to remove math notation that is specific to that paper. However, this is an interesting problem. Intuitively, in soft selection the fitness of an individual is indeed a function of group membership. After all it is the phenotype of the individual relative to the population mean that determines its fitness, with an intermediate phenotype individual having a high fitness in a group of low phenotype individuals, and a low fitness in a group of high phenotype individuals.

So, what is going on? I think the easiest thing to do is to do some math. Further, to remove the politics, lets not think about contextual analysis. Instead, think of selection on two correlated traits, say body length, and body weight. These we can imagine to be correlated because generally longer animals are also bigger. In this population we can imagine that the population has a mean length of 60 inches and a mean weight of 100 pounds. Further, there is a correlation between length and weight of 0.5, giving a phenotypic covariance matrix of (Phenotypic NOT genotypic covariance matrix. Also, keeping it as simple as possible):

Soft Selection eq 1

So, now the problem. Because these traits are correlated, there is not only direct selection, that is if you select on length, length will change, but also indirect selection: Selecting on weight will also change body length. It turns out we can separate the effects of selection acting directly on a trait (direct selection) from selection acting on another correlated trait (indirect selection). The vector that does this is called the selection gradient, β. It is just like the selection vector, except that it mathematically removes the effects of indirect selection. Thus, if we have selection only acting on body length then the selection gradient might look like this (again keeping it simple)

Soft Selection eq 2

Indicating a selection strength of 1 on Length, and a selection strength of 0 on weight. Of course, we are asserting this direct selection, but what we actually observe is S, the selection vector, which includes both the effects of direct AND indirect selection. So what would S look like? To do determine this we need to recognize that the gradient is actually equal to β = P-1S, where P-1 is the inverse of the phenotypic covariance matrix:

Soft Selection eq 3

Then doing a bunch of algebra nobody needs to know we can solve for S, and discover that:

Soft Selection eq 4

In other words, if we apply selection only on length because the traits are correlated we will also see a change in weight.

OK, Now contextual analysis and soft selection. In the case of soft selection by assumption we have:

Soft Selection eq 5

In other words, we are assuming that there is no covariance between the group mean trait and group mean fitness. I am running out of space, but basically if we use only simple covariances:

Soft Selection eq 6

But, of course, what is important is not S, but β. Thus, if we want to calculate the selection gradient we need to calculate β:

Soft Selection eq 7

The point is that in order to have no variation in fitness among groups you need to have selection at the group level to remove that variation in fitness.

I could have made the same story with the original example of length and weight. If you want to select for longer animals without changing their weight you will need to select against weight. That is, you will need to select for long skinny animals. In other words, lack of covariance between fitness and a trait correlated with one under selection does not come for free. You need to actively apply selection to remove that covariation.

Thus, the point of this whole story is that the lack of covariance between group mean fitness and the group trait is NOT evidence for lack of group selection. Far from it, it is in fact evidence that group selection is acting. To believe otherwise is to simply not understand how selection on correlated characters works.

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